Smale's 6th problem for generic masses

We give a new method to attempt to prove that, for a given $n$, there are finitely many equivalence classes of planar central configurations in the Newtonian $n$-body problem for generic masses. The human part of the proof relies on tropical geometry. The crux of our technique is in a computation that we have completed for $n\le 5$, thus confirming the celebrated result of Albouy and Kaloshin.